Physical meaning and well-posedness of grand canonical ensembles with imaginary chemical potentials

Establish the physical interpretation and mathematical well-posedness, including convergence properties, of the grand canonical ensemble defined by the density matrix ρ = exp[−β(H − Φ_E Q − Ω_E J)] that arises from analytic continuation in the Euclidean treatment of rotating, charged AdS black holes, clarifying whether such ensembles correspond to meaningful observables and under what conditions they converge.

Background

In the discussion of rotating, charged AdS black holes, rendering the metric Riemannian by analytically continuing the rotation and charge parameters leads to a density matrix with imaginary chemical potentials. While semiclassical computations using this continuation reproduce sensible thermodynamic relations, the ensemble’s physical meaning and mathematical soundness are questioned.

A resolution would require a clear interpretation of observables computed with such density matrices and rigorous criteria ensuring convergence, potentially tied to constraints on allowed complex saddles or boundary conditions.